An Evolutionary Game-Theoretic Perspective on the Prisoner’s Dilemma

Abstract

The Prisoner’s Dilemma is the canonical model in microeconomics for the tension between individual rationality and collective welfare. Standard game-theoretic analysis predicts that mutual defection is the unique Nash equilibrium, despite being Pareto-dominated by mutual cooperation. This study revisits the Prisoner’s Dilemma through the lens of evolutionary game theory, in which strategy frequencies in a population evolve according to differential payoffs. We derive the replicator equation for the one-shot game and show analytically that the all-defectors state is the unique asymptotically stable equilibrium and the unique evolutionarily stable strategy. We then extend the analysis to the infinitely repeated game with continuation probability , contrasting the reactive strategy Tit-for-Tat (TFT) with Always Defect (ALLD). We derive the critical continuation probability  above which TFT becomes an ESS within the restricted two-strategy space (that is, resists invasion by ALLD), the bistable replicator dynamics that emerge in that regime, and a closed-form expression for the unstable interior equilibrium  that separates the two basins of attraction. The bifurcation structure is interpreted economically as a precise formalisation of Axelrod’s “shadow of the future.” All results are derived from first principles and illustrated with numerical examples.

Keywords

• Evolutionary Game Theory
• Evolutionary Stable Strategy
• Prisoner’s Dilemma
• Repeated Games

Mahkûm İkilemine Evrimsel Oyun Teorik Bir Bakış

Abstract

Mahkûm İkilemi, mikro ekonomide bireysel rasyonellik ile toplumsal refah arasındaki gerilimi modelleyen kanonik bir yapıdır. Standart oyun-teorik analiz, karşılıklı işbirliği tarafından Pareto baskılanmasına rağmen karşılıklı ihanetin tek Nash dengesi olduğunu öngörmektedir. Bu çalışma, mahkûm ikilemini popülasyondaki strateji frekanslarının farklı kazanımlara bağlı olarak evrildiği evrimsel oyun teorisi perspektifinden yeniden ele almaktadır. Tek seferlik oyun için replikatör denklemi türetilmekte ve "tümü-ihanet" durumunun yegâne asimptotik kararlı denge ve yegâne evrimsel kararlı strateji olduğu analitik olarak gösterilmektedir. Analiz, devam etme olasılığı   ile sonsuz tekrarlı oyunu genişletilerek, tepkisel strateji kısasa kısas ile her zaman ihanet stratejilerini karşılaştırmaktadır. Kısasa kısasın kısıtlanmış iki stratejili uzayda (yani her zaman ihanet stratejisinin istilasına karşı) evrimsel durağan strateji hâline gelmesi için kritik devam etme olasılığı   türetilmekte, bu rejimde ortaya çıkan iki kararlı dengeli (bistable) replikatör dinamikleri incelenmekte ve iki çekim havzasını ayıran kararsız iç denge   için kapalı-form bir ifade elde edilmektedir. Çatallaşma yapısı, Axelrod'un "geleceğin gölgesi" kavramının kesin bir biçimselleştirmesi olarak ekonomik açıdan yorumlanmaktadır. Tüm sonuçlar ilk ilkelerden türetilmekte ve sayısal örneklerle desteklenmektedir.

Keywords

• Evrimsel Oyun Teorisi
• Evrimsel Durağan Strateji
• Mahkûmlar İkilemi
• Tekrarlı Oyunlar

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